Quite simply, $$\frac{e^{x}-1}{e^{x/2}-1}=e^{x/2}+1$$ I have no idea what combination of exponential properties or fraction identity tricks would result in the RHS of the equation. I apologize if it's super obvious and I'm just not seeing it.
2026-03-26 17:28:31.1774546111
Exponential sum divided by exponential sum
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If $u=e^{x/2}$ then the top is $u^2-1$ and bottom is $u-1.$ Now use $u^2-1=(u-1)(u+1).$